The von Neumann entropy of a quantum state is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state $\rho$ can be formulated as the difference of the number of physical qubits and the "relative entropy distance" between $\rho$ and the maximally mixed state, here we define the entropy of a channel $\mathcal{N}$ as the difference of the number of physical qubits of the channel output with the "relative entropy distance" between $\mathcal{N}$ and the completely depolarizing channel. We prove that this definition satisfies all of the axioms, recently put forward in [Gour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. The entropy of a channel can be negative for certain channels, but this negativity has an operational interpretation in terms of the channel merging protocol. We define Renyi and min-entropies of a channel and prove that they satisfy the axioms required for a channel entropy function. Among other results, we also prove that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.
翻译:量子状态的 von Neumann engropy 是物理和信息理论中的核心概念, 具有许多令人信服的物理解释。 有一种观点认为, 量子力学中最基本的概念是量子频道, 因为量子系的量子状态、 单一进化、 测量和丢弃, 可以被视为一定的量子频道。 因此, 一个重要的目标是定义一个一致和有意义的量子频道的进化概念。 受以下事实的启发: 州 $\rho$ 的进化可被配成物理量子数和“ 反向导” 之间的差别。 量子力力力力力学中, 量子力学中, 量子流中, 磁力轨道中, 流流中, 磁力中, 流流中, 磁力中, 磁力中, 磁力中, 磁力导中, 流流流中, 磁力中, 磁力中, 流中, 磁力中, 流中, 流中, 流流流流中, 流流中, 流中, 需要一个磁力中, 流中, 磁力中, 磁力中, 流中, 流中, 流中, 流中, 磁力, 流中, 流中, 流中, 流中, 磁力函数中, 流中, 流中, 流流函数中,,,, 流函数中, 流函数中,, 流函数中, 流函数中, 流函数。